Problems on Isosceles Triangles with Detailed Solutions

Isosceles Triangle Formulas

An Isosceles triangle has two equal sides with the angles opposite to them equal. The relationship between the lateral side \( a \), the based \( b \) of the isosceles triangle, its area A, height h, inscribed and circumscribed radii r and R respectively are give by:

Problem 1

What is the area of an isosceles triangle with base b of 8 cm and lateral a side 5 cm?

Problem 2

What is the base of an isosceles triangle with lateral side a = 5 cm and area 6 cm 2?

Problem 3

What is the lateral side of an isosceles triangle with area 20 unit 2 and base 10 units?

Problem 4

What is the lateral side of an isosceles triangle such that its height h ( perpendicular to its base b) is 4 cm shorter than its base b and its area is 30 cm 2?

Problem 5

ABC and BCD are isosceles triangles. Find the size of angle BDE.

Problem 6

ABC and CDE are isosceles triangles. Find the size of angle CED.

Problem 7

Find the area of the circle inscribed to an isosceles triangle of base 10 units and lateral side 12 units.

Problem 8

Find the ratio of the radii of the circumscribed and inscribed circles to an isosceles triangle of base b units and lateral side a units such that a = 2 b.

Problem 9

Find the lateral side and base of an isosceles triangle whose height ( perpendicular to the base ) is 16 cm and the radius of its circumscribed circle is 9 cm.

Problem 10

What is the area of an isosceles triangle of lateral side 2 units that is similar to another isosceles triangle of lateral side 10 units and base 12 units?

Solutions to the Above Questions

1. Solution
   
   Apply Pythagora’s theorem to the right triangle CC’B (see figure at top) to write
   
   a 2 = (b/2) 2 + h 2
   h = a 2 – (b/2) 2 = 5 2 – 4 2 which gives h = 3
   
   Area A = (1/2) b h = (1/2) 8�3 = 12 unit 2
2. Solution
   
   Use formula of area of isosceles triangle
   
   A = (1/2) a 2 sin(α)
   
   to find α as follows
   
   sin(α) = 2 A / a 2 = 2 * 6 / 5 2 = 12 / 25
   
   α = arcsin(12 / 25)
   
   Using right triangle CC’B (see figure at top), we can write
   
   sin(α/2) = (b/2) / a = b / 2a
   
   b = 2 a sin (α/2)
   
   b = 2 a sin( (1/2) arcsin(12 / 25) ) = 10 sin( (1/2) arcsin(12 / 25) ) ≅ 2.48
3. Solution
   
   Use formula of area of isosceles triangle to write
   
   A = (1/2) b h = 20
   
   Given b = 10, find h
   
   h = 40 / 10 = 4
   
   Pythagora’s theorem used in the right triangle CC’B (see figure at top) to write
   
   a 2 = (b/2) 2 + h 2 = √ ( 5 2 + 4 2) = √41
4. Solution
   
   Use formula of area of isosceles triangle to write
   
   A = (1/2) b h = 30
   
   h = b – 4 (given)
   
   Substitute h by b – 4 in the formula for A
   
   A = (1/2) b (b – 4) = 30
   
   Gives the equation: b 2 – 4 b – 60 = 0
   
   Solutions to the equation: b = 10 and b = – 6
   
   b is a length and therefore is positive b = 10 , h = b – 4 = 6
   
   a = √ (h 2 + (b/2) 2) = √ (36 + 25) = √61
5. Solution
   
   ABC is an isosceles triangle and therefore
   
   ∠CAB = ∠ABC
   
   Also: ∠CAB + ∠ABC = 180 – 66
   
   2 ∠ABC = 180 – 66
   
   gives: ∠ABC = 57�
   
   BCD is a right isosceles triangle; hence
   
   ∠CBD = ∠CDB = (180 – 90)/2 = 45�
   
   Note that ∠ABC, ∠CBD and ∠DBE make a straight angle. Hence
   
   ∠ABC + ∠CBD + ∠DBE = 180�
   
   gives ∠DBE = 180 – ∠ABC – ∠CBD = 180 – 57 – 45 = 78�
   
   DBE is a right triangle ; hence
   
   ∠BDE = 90 – 78 = 12�
6. Solution
   
   Straight angle at B hence
   
   ∠ABC = 180 – 116 = 64�
   
   ABC is an isosceles triangle and therefore
   
   ∠CAB = ∠ABC = 64�
   
   In trangle ABC: ∠BCA + ∠CAB + ∠ABC = 180�
   
   Hence: ∠BCA = 180 – 64 – 64 = 52�
   
   ∠DCE = ∠ BCA = 52�
   
   CDE isosceles: ∠DCE = ∠CDE = 52�
   
   In triangle CDE we have: ∠DCE + ∠CDE + ∠CED = 180�
   
   ∠CED = 180 – 52 – 52 = 76�
7. Solution
   
   Radius of inscribed circle to an isosceles triangle of base b = 10 and lateral side a = 12 is given by
   
   r = \( \dfrac{b}{2} \sqrt{\dfrac{2a-b}{2a+b}} = 5 \sqrt{\dfrac{4}{34}} \)
   
   Area A of circle of radius r is given by: A = π r 2 = 100π / 34
8. Solution
   
   Radius R of the circumscibed circle and radius r of the inscribed circle to the same isosceles triangle of base b and lateral side a are given by
   
   R = \( \dfrac{a^2}{\sqrt{4a^2-b^2}} \) and r = \( \dfrac{b}{2} \sqrt{\dfrac{2a-b}{2a+b}} \)
   
   Substitute a by 2 b (given) in both formulas and simplify
   
   R = \( \dfrac{(2b)^2}{\sqrt{4(2b)^2-b^2}} = \dfrac{4b^2}{\sqrt{15b^2}} =\dfrac{4b}{\sqrt15} \)
   
   r = \( \dfrac{b}{2} \sqrt{\dfrac{4b-b}{4b+b}} = (b/2) \sqrt{\dfrac{3}{5}} \)
   
   Simplify ratio: R / r = 8/3
9. Solution
   
   Radius R of the circumscibed circle an isosceles triangle of base b and lateral side a are given by
   
   R = \( \dfrac{a^2}{\sqrt{4a^2-b^2}} \)
   
   Relationship between h, b and a (Pythagora in triangle CC’B see figure above)
   
   a 2 = (b/2) 2 + h 2
   Gives: a 2 = b 2 / 4 + h 2 or 4 h 2 = 4 a 2 – b 2
   Substitute R by 9 (given) and 4 a 2 – b 2 by 4 h 2 in the formula for R above to get the equation
   
   9 = \( \dfrac{a^2}{\sqrt{4h^2}} = a^2 / (2h) \)
   
   a 2 = 9 � 2 � 16
   
   gives a = 12 √2 cm.
10. Solution
    
    Let A1 and A2 be the areas of the triangle with lateral sides a1 = 2 and a2 = 10 respectively.
    
    Formulas for A1 and A2
    A1 = (1 / 2) a1 2sin(α) and A2 = (1 / 2) a2 2 sin(α) , corresponding angles are equal in similar triangles.
    
    Hence the ratio: A1 / A2 = (1 / 2) a1 2sin(α) / (1 / 2)a2 2sin(α) = a1 2 / a2 2
    We now need to find A2
    Use Pythagora’s theorem in the right triangle CC’B (see figure at top) to write
    
    a 2 = (b/2) 2 + h 2
    h = a 2 – (b/2) 2 = 10 2 – 6 2 which gives h = 8
    
    Area A2 = (1 / 2) b h = (1 / 2) 12�8 = 48 unit 2
    a1 2 / a2 2 = A1 / A2
    A1 = A2 � a1 2 / a2 2 = 48 � 2 2/ 10 2 = 1.92 unit 2